\(\def\N{\mathbb{N}}\) \(\def\Z{\mathbb{Z}}\) \(\def\Q{\mathbb{Q}}\) \(\def\R{\mathbb{R}}\) \(\def\C{\mathbb{C}}\) \(\def\H{\mathbb{H}}\) \(\def\6{\partial}\) \(\DeclareMathOperator\Res{Res}\) \(\DeclareMathOperator\M{M}\) \(\DeclareMathOperator\ord{ord}\) \(\DeclareMathOperator\const{const}\) \(\DeclareMathOperator{\arccosh}{arccosh}\) \(\DeclareMathOperator{\arcsinh}{arcsinh}\) \(\DeclareMathOperator\id{id}\) \(\DeclareMathOperator\rk{rk}\) \(\DeclareMathOperator\tr{tr}\) \(\def\pt{\mathrm{pt}}\) \(\DeclareMathOperator\colim{colim}\) \(\DeclareMathOperator\Hom{Hom}\) \(\DeclareMathOperator\End{End}\) \(\DeclareMathOperator\Aut{Aut}\) \(\let\Im\relax\DeclareMathOperator\Im{Im}\) \(\let\Re\relax\DeclareMathOperator\Re{Re}\) \(\DeclareMathOperator\Ker{Ker}\) \(\DeclareMathOperator\Coker{Coker}\) \(\DeclareMathOperator\Map{Map}\) \(\def\GL{\mathrm{GL}}\) \(\def\SL{\mathrm{SL}}\) \(\def\O{\mathrm{O}}\) \(\def\SO{\mathrm{SO}}\) \(\def\Spin{\mathrm{Spin}}\) \(\def\U{\mathrm{U}}\) \(\def\SU{\mathrm{SU}}\) \(\def\g{{\mathfrak g}}\) \(\def\h{{\mathfrak h}}\) \(\def\gl{{\mathfrak{gl}}}\) \(\def\sl{{\mathfrak{sl}}}\) \(\def\sp{{\mathfrak{sp}}}\) \(\def\so{{\mathfrak{so}}}\) \(\def\spin{{\mathfrak{spin}}}\) \(\def\u{{\mathfrak u}}\) \(\def\su{{\mathfrak{su}}}\) \(\def\cA{\mathcal{A}}\) \(\def\cB{\mathcal{B}}\) \(\def\cC{\mathcal{C}}\) \(\def\cD{\mathcal{D}}\) \(\def\cE{\mathcal{E}}\) \(\def\cF{\mathcal{F}}\) \(\def\cG{\mathcal{G}}\) \(\def\cH{\mathcal{H}}\) \(\def\cI{\mathcal{I}}\) \(\def\cJ{\mathcal{J}}\) \(\def\cK{\mathcal{K}}\) \(\def\cL{\mathcal{L}}\) \(\def\cM{\mathcal{M}}\) \(\def\cN{\mathcal{N}}\) \(\def\cO{\mathcal{O}}\) \(\def\cP{\mathcal{P}}\) \(\def\cQ{\mathcal{Q}}\) \(\def\cR{\mathcal{R}}\) \(\def\cS{\mathcal{S}}\) \(\def\cT{\mathcal{T}}\) \(\def\cU{\mathcal{U}}\) \(\def\cV{\mathcal{V}}\) \(\def\cW{\mathcal{W}}\) \(\def\cX{\mathcal{X}}\) \(\def\cY{\mathcal{Y}}\) \(\def\cZ{\mathcal{Z}}\) \(\def\al{\alpha}\) \(\def\be{\beta}\) \(\def\ga{\gamma}\) \(\def\de{\delta}\) \(\def\ep{\epsilon}\) \(\def\ze{\zeta}\) \(\def\th{\theta}\) \(\def\io{\iota}\) \(\def\ka{\kappa}\) \(\def\la{\lambda}\) \(\def\si{\sigma}\) \(\def\up{\upsilon}\) \(\def\vp{\varphi}\) \(\def\om{\omega}\) \(\def\De{\Delta}\) \(\def\Ka{{\rm K}}\) \(\def\La{\Lambda}\) \(\def\Om{\Omega}\) \(\def\Ga{\Gamma}\) \(\def\Si{\Sigma}\) \(\def\Th{\Theta}\) \(\def\Up{\Upsilon}\) \(\def\Chi{{\rm X}}\) \(\def\Tau{{T}}\) \(\def\Nu{{\rm N}}\) \(\def\op{\oplus}\) \(\def\ot{\otimes}\) \(\def\t{\times}\) \(\def\bt{\boxtimes}\) \(\def\bu{\bullet}\) \(\def\iy{\infty}\) \(\def\longra{\longrightarrow}\) \(\def\an#1{\langle #1 \rangle}\) \(\def\ban#1{\bigl\langle #1 \bigr\rangle}\) \(\def\llbracket{{\normalsize\unicode{x27E6}}} \def\rrbracket{{\normalsize\unicode{x27E7}}} \) \(\def\lb{\llbracket}\) \(\def\rb{\rrbracket}\) \(\def\ul{\underline}\) \(\def\ol{\overline}\)

7  Cauchy’s theorem

Definition 7.1 Let \(D\subset\C\) and \(\ga_0, \ga_1\colon[a,b]\to D\) curves in \(D.\)

  1. A homotopy in \(D\) between paths \(\ga_0, \ga_1\) is a continuous map \[\Ga\colon[0,1]\t[a,b]\longra D, (s,t)\longmapsto\Ga_s(t),\] such that \(\Ga_0(t)=\ga_0(t)\) and \(\Ga_1(t)=\ga_1(t)\) for all \(t\in[a,b].\) Then \(\ga_0, \ga_1\) are called (freely) homotopic paths in \(D.\)

    If, additionally, \(\Ga_s(a)=p\) and \(\Ga_s(b)=q\) are constant in \(s\in[0,1],\) we call \(\Ga\) a path homotopy in \(D\) and \(\ga_0, \ga_1\) path-homotopic in \(D\).

  2. If \(\ga_0, \ga_1\) are loops, a homotopy of loops in \(D\) is a homotopy \(\Ga\) in \(D\) with the additional property that \(\Ga_s\) is a loop for each \(s\in[0,1].\) Then \(\ga_0, \ga_1\) are called (freely) homotopic loops in \(D.\)

    A loop is null-homotopic in \(D\) if there is a homotopy of loops in \(D\) to the constant loop.

A path homotopy is a video of paths with fixed endpoints.

A path homotopy is a video of paths with fixed endpoints.

Remark 7.1. In the following we will also suppose that \(\Ga\) is piecewise C1, meaning there exist subdivisions \[0=s_0<s_1<\cdots<s_m=1,\qquad a=t_0<t_1<\cdots<t_n=b\]

such that each restriction \(\Ga|_{[s_{j-1},s_j]\t[t_{k-1},t_k]}\) is continuously differentiable.

Example 7.1  

A set \(D\) is star-shaped if there exists a focal point \(z_0\in D\) such that for each \(z\in D\) the straight line segment \(tz+(1-t)z_0,\) \(t\in[0,1],\) is contained in \(D.\) Disks, the complex plane, and rectangles are star-shaped. Every loop \(\ga\) in a star-shaped domain is null-homotopic. If the focal point agrees with the base of the loop, the path homotopy is

\[\Ga_s(t) = (1-t)\ga(t) + tz_0. \tag{7.1}\]

In general, the path homotopy is more complicated to write down. We omit it, since we will not need this fact below.

Theorem 7.1 (Cauchy’s theorem) Let \(f\colon U\to \C\) be a holomorphic function on an open set. Let \(\ga_0, \ga_1\) be piecewise C1 curves in \(U\) that are path-homotopic in \(U.\) Then

\[\int_{\ga_0}f(z)dz = \int_{\ga_1}f(z)dz.\]

Proof.

Pick a path homotopy \(\Ga\colon[0,1]\t[a,b]\to U.\) By reparameterizing we may assume \([a,b]=[0,1].\) Write \(R^{(0)}=[0,1]\t[0,1]\) for the domain of \(\Ga\) and let \(\6 R^{(0)}\) be its piecewise linear boundary path, with the obvious counterclockwise parameterization (see Figure 7.2).

Path homotopy when $\ga_0,$ $\ga_1$ are loops.
Figure 7.2: Path homotopy when \(\ga_0,\) \(\ga_1\) are loops.

The boundary path has four segments and the path integral along each constant path vanishes. Using Proposition 6.2(b),(c), we then find that \[\int_{\ga_0} f(z)dz-\int_{\ga_1} f(z)dz=\int_{\Ga(\6 R^{(0)})} f(z)dz.\]

Subdivide \(R^{(0)}\) into four congruent rectangles \(R^{(0)}_i\) as in Figure 7.3. Using the fact that the path integrals in opposite directions cancel, we find \[\left|\int_{\Ga(\6 R^{(0)})} f(z)dz\right| = \left|\sum_{i=1}^4 \int_{\Ga(\6 R^{(0)}_i)} f(z)dz\right|\leqslant \sum_{i=1}^4 \left|\int_{\Ga(\6 R^{(0)}_i)} f(z)dz\right|.\]

Subdivision and cancellation of opposite paths
Figure 7.3: Subdivision and cancellation of opposite paths

Let \(R^{(1)}\) be the rectangle \(R_i^{(0)}\) for which \(\left|\int_{\Ga(\6 R^{(0)}_i)} f(z)dz\right|\) is maximal. Then \[\left|\int_{\Ga(\6 R^{(0)})} f(z)dz\right|\leqslant4\left|\int_{\Ga(\6 R^{(1)})} f(z)dz\right|.\]

Now repeat this process with \(R^{(1)}\) to obtain \(R^{(2)}\) and so forth. This yields a sequence of rectangles \(R^{(n)}\) as in Figure 7.4 with sides of length \(2^{-n}\) and \[\left|\int_{\Ga(\6 R^{(0)})} f(z)dz\right|\leqslant4^n\left|\int_{\Ga(\6 R^{(n)})} f(z)dz\right|. \tag{7.2}\]

Convergent sequence of rectangles $\bigcap R^{(n)}={(s_0,t_0)}$
Figure 7.4: Convergent sequence of rectangles \(\bigcap R^{(n)}=\{(s_0,t_0)\}\)

As the side lengths tend to zero, the midpoints \((s^{(n)},t^{(n)})\) of the rectangles \(R^{(n)}\) are a Cauchy sequence, so converge to some limit \((s_0,t_0).\) Let \(z_0=\Ga_{s_0}(t_0).\) Since \(f\) is complex differentiable at \(z_0,\) we may write \[f(z)=f(z_0) + (z-z_0)f'(z_0) + (z-z_0)\rho(z),\qquad\lim_{z\to z_0}\rho(z)=0. \tag{7.3}\] By Example 6.4,\[\int_{\Ga(\6 R^{(n)})}\left(f(z_0) + (z-z_0)f'(z_0)\right)dz=0. \tag{7.4}\]

As \(\Ga\) is piecewise C1, its derivative is bounded in norm by some \(C>0.\) Let \(\ep>0.\) Pick \(n_0\in\N\) such that for all \(n\geqslant n_0\) and \(z\in R^{(n)}\) we have \(|\rho(z)|<\ep.\) As the side lengths of \(R^{(n)}\) are \(2^{-n}\) and \(z_0\in R^{(n)},\) we have \(|z-z_0|\leqslant\sqrt{2}2^{-n}\) for all \(z\in\6 R^{(n)}.\) Combining this with Equation 7.3 and Equation 7.4, we can estimate

\[\begin{align*} \left|\int_{\Ga(\6 R^{(n)})} f(z)dz\right|&= \left|\int_{\Ga(\6 R^{(n)})}(z-z_0)\rho(z)dz\right|\\ &\leqslant L(\Ga(\6 R^{(n)}))\sqrt{2}2^{-n}\ep = 4C\cdot 2^{-n}\sqrt{2}2^{-n}\ep \end{align*}\]

The last inequality follows by Equation 6.7.

Hence \[\left|\int_\ga f(z)dz\right|=\left|\int_{\Ga(\6 R^{(0)})} f(z)dz\right|\overset{}{\leqslant}4C\sqrt{2}\ep.\] The final inequality follows from Equation 7.2.

As \(\ep>0\) is arbitrary, the left hand side must be zero.

Theorem 7.2 Let \(\ga_0, \ga_1\) be piecewise C1 loops in \(U\) that are freely homotopic in \(U.\) Suppose \(f\colon U\to\C\) is a holomorphic function. Then

\[\int_{\ga_0}f(z)dz = \int_{\ga_1}f(z)dz.\]In particular, if \(\ga\) is a loop that is homotopic in \(U\) to the constant loop, then\[\int_\ga f(z)dz=0. \tag{7.5}\]

Proof.

Let \(\Ga\) be the homotopy between \(\ga_0\) and \(\ga_1.\) Let \(\eta(s)=\Ga_s(0).\) Figure 7.5 describes the construction of a path homotopy between \(\eta\ast\ga_0\ast(-\eta)\) and \(\ga_1.\)

Construction of path homotopy
Figure 7.5: Construction of path homotopy

Hence Theorem 7.1 implies our claim, using the fact that the path integrals over \(\eta\) and \(-\eta\) cancel each other.

Questions for further discussion

  • How should Figure 7.5 be interpreted? Can you give a formula for the path homotopy (consider cases)?
  • Give a counterexample to Cauchy’s theorem when (i) \(\ga\) is not null-homotopic (ii) \(f(z)\) is not holomorphic

  • Does Cauchy’s theorem hold for \(f(z)=\ol{z}\)?

7.1 Exercises

Exercise 7.1
  1. Sketch the curves \[\begin{align*} \ga_0\colon[-1,1]&\longra\C,&\ga_0(t)&=t,\\ \ga_1\colon[-1,1]&\longra\C,&\ga_1(t)&=e^{i\pi\frac{1-t}{2}} \end{align*}\] and show that they are path-homotopic.

    1. Let \(0<r_0<r_1\) and \(z_0\in\C.\) Prove that the loops \(\partial D_{r_0}(z_0)\) and \(\partial D_{r_1}(z_0)\) are freely homotopic in the closed annulus \(\ol A_{r_0,r_1}(z_0).\)
Exercise 7.2

Let \(\al+i\be\in\C.\) Determine \[\int_a^b e^{(\al+i\be)t} dt\] to compute \[\int_a^b e^{\al t}\cos(\be t)dt.\]

Exercise 7.3

Let \(\C^-=\C\setminus(-\iy,0]\) be the slit plane.

  1. Show that any two points in \(\C^-\) may be connected by a path in \(\C^-.\) Hence \(\C^-\) is path-connected.
  2. Show that every closed curve \(\ga\colon[a,b]\to\C^-\) is null-homotopic in \(\C^-\) by finding a homotopy \[\Ga\colon[0,1]\t[a,b]\longra\C^-, (s,t)\longmapsto\Ga_s(t)\] satisfying \(\Ga_0(t)=\ga(t)\) and \(\Ga_1(t)=1\) for all \(t\in[a,b].\) Hence \(\C^-\) is simply-connected.
  3. Prove the analogues of a. and b. for a disk \(D_r(z_0).\)
  4. Use Cauchy’s Theorem to prove that the punctured plane \(\C^\t=\C\setminus\{0\}\) is not simply-connected. That is, there exists a closed curve in \(\C^\t\) that is not null-homotopic in \(\C^\t.\)
Exercise 7.4

Let \(0<b<1.\)

  1. Using the geometric series, find the power series expansion \[\frac{1}{z-1/b}=\sum_{n=0}^\iy a_n(z-b)^n\] with center \(z_0=b\) and determine the radius of convergence \(\rho.\)
  2. Use a. to show that for all \(0<r<\rho\) \[\int_{\partial D_r(b)}\frac{dz}{(z-b)(z-1/b)}=\frac{2\pi i}{b-1/b}.\]
  3. Use c. to compute \[\int_0^{2\pi}\frac{dt}{1-2b\cos(t)+b^2}.\]
Exercise 7.5

Let \(\ga\colon[0,1]\to D\) be a curve in \(D\subset\C\) and let \(-\ga\colon[0,1]\to D,\) \((-\ga)(t)=\ga(1-t)\) be the opposite curve. Prove that \(\ga\ast(-\ga)\) is path-homotopic to a constant loop.